We tackle the problem of accelerating column generation (CG) approaches to set cover formulations in operations research. At each iteration of CG we generate a dual solution that approximately solves the LP over all columns consisting of a subset of columns in the nascent set. We refer to this linear program (LP) as the Family Restricted Master Problem (FRMP), which provides a tighter bound on the master problem at each iteration of CG, while preserving efficient inference. For example, in the single source capacitated facility location problem (SSCFLP) the family of a column $l$ associated with facility $f$ and customer set $N_l$ contains the set of columns associated with $f$ and the customer set that lies in the power set of $N_l$. The solution to FRMP optimization is attacked with a coordinate ascent method in the dual. The generation of direction of travel corresponds to solving the restricted master problem over columns corresponding to the reduced lowest cost column in each family given specific dual variables based on the incumbent dual, and is easily generated without resolving complex pricing problems. We apply our algorithm to the SSCFLP and demonstrate improved performance over two relevant baselines.